Understanding the motion of macromolecules in topologically restrictive environments is a problem of fundamental and practical interest and has been an ongoing challenge for over a half century.
In a concentrated solution or melt consisting of sufficiently long polymer chains, the motion of a test chain becomes restricted at a length scale \(d_T\)  (the tube diameter) due to interchain uncrossability, which forces the test chain to diffuse along a curvilinear tube-like path (the primitive path).
The motion of a test chain along the tube is presumed to proceed via the same dynamics that govern the motion of the chain on length scales below \(d_T\)  [1]}, [2]}.
This molecular picture for the motion of a topologically constrained polymer was provided by de Gennes, Doi and Edwards [3]}, [1]} and motivated the highly successful [5]}, [6]} phenomenological Doi-Edwards (DE) tube model [7]}, , ,  (and its modern adaptations [8]}, [9]}) for entangled polymer dynamics and rheology.
